Satisfying Dirichlet Boundary Conditions Research Articles

Global exact controllability of the viscous and resistive MHD system in a rectangle thanks to the lateral sides and to distributed phantom forces

We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.

Casimir effect in 2+1 Hořava gravity

We study the Casimir effect of a membrane embedded in 2+1 dimensions flat cone generated by a massive particle located at the origin of the coordinate system. The flat cone is an exact solution of the nonprojectable Hořava theory, similar to general relativity. We consider a scalar field satisfying Dirichlet boundary conditions, and regularize the spectrum using the ζ–function technique. In addition, we include the effects of temperature in our analysis. Our results show that the Casimir force depends on three factors: the anisotropic scaling z, the mass of the point particle, and the temperature.

Physics-informed neural networks for mesh deformation with exact boundary enforcement

In this work, we have applied physics-informed neural networks (PINN) for solving mesh deformation problems. We used the collocation PINN method to capture the new positions of the vertex nodes while preserving the connectivity information. We use linear elasticity equations for mesh deformation. To prevent vertex collisions or edge overlap, the mesh movement in this work is conducted in steps with relatively small movements. For moving boundary problems, the exact position of the boundary is essential for having an accurate solution. However, PINNs are frequently unable to satisfy Dirichlet boundary conditions exactly. To overcome this issue, we have used hard boundary condition enforcement to automatically satisfy Dirichlet boundary conditions. Specifically, we first trained a PINN with soft boundary conditions to obtain a particular solution. Then, this solution was tuned with exact boundary positions and a proper distance function by using a new PINN considering only the equation residual. To assess the accuracy of our approach, we used the classical translation and rotation tests and compared them with a proper mesh quality metric considering the change in the element area and shape. The results show the accuracy of this approach is comparable with that of finite element solutions. We also solved different moving boundary problems, resembling commonly used fluid–structure interaction problems. This work provides insight into using PINN for mesh-deformation problems without needing a discretization scheme with reasonable accuracy.

Glue-on AdS holography for Toverline{T} -deformed CFTs

The Toverline{T} deformation is a solvable irrelevant deformation whose properties depend on the sign of the deformation parameter μ. In particular, Toverline{T} -deformed CFTs with μ < 0 have been proposed to be holographically dual to Einstein gravity where the metric satisfies Dirichlet boundary conditions at a finite cutoff surface. In this paper, we put forward a holographic proposal for Toverline{T} -deformed CFTs with μ > 0, in which case the bulk geometry is constructed by gluing a patch of AdS3 to the original spacetime. As evidence, we show that the Toverline{T} trace flow equation, the spectrum on the cylinder, and the partition function on the torus and the sphere, among other results, can all be reproduced from bulk calculations in glue-on AdS3.

A remark on quantitative unique continuation from subsets of the boundary of positive measure

The question of uniqueness of harmonic functions in a domain Ω ⊂ R d \Omega \subset \mathbb {R}^d with boundary ∂ Ω \partial \Omega , satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset of the boundary ω \omega of positive d − 1 d-1 dimensional measure has attracted a lot of attention (see e.g. V. Adolfsson and L. Escauriaza [Comm. Pure Appl. Math. 50 (1997), pp. 935–969]; F.-H. Lin [Comm. Pure Appl. Math. 44 (1991), pp. 287–308]; X. Tolsa [Comm. Pure Appl. Math. 76 (2023), pp. 305–336]). It is essentially a consequence of Carleman estimates when ω \omega contains an open subset of the boundary (uniqueness for second order elliptic operators across an hypersurface). The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, Birkhäuser/Springer, Cham, 2018, pp. 333-344; Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific Publishing, Hackensack, NJ, 2018, pp. 2391–2411], we consider the simpler case of W 2 , ∞ W^{2, \infty } domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2 , ∞ W^{2, \infty } domain with boundary.

Data assimilation with higher order finite element interpolants

AbstractThe efficiency of a nudging data assimilation algorithm using higher order finite element interpolating operators is studied. Numerical experiments are presented for the 2D Navier–Stokes equations in three cases: shear flow in an annulus, a forced flow in a disk with an off‐center cavity, and a forced flow in a box all satisfying Dirichlet boundary conditions. In all three cases, second order interpolation of coarse‐grain data is shown to outperform first order interpolation. Convergence of the nudged solution to that of a direct numerical reference solution is proved. The analysis points to a trade‐off in the estimates for higher order interpolating operators

Nontrivial solutions for a nonlinear νth order Atıcı-Eloe fractional difference equation satisfying Dirichlet boundary conditions

Nontrivial solutions for a nonlinear νth order Atıcı-Eloe fractional difference equation satisfying Dirichlet boundary conditions

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artificial neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ(x), an approximate distance function to the boundary of a domain in Rd. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ(x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

On the abundance of supersymmetric strings in AdS3 × S 3 × S 3 × S 1 describing BPS line operators

We study supersymmetric open strings in type IIB AdS3 × S 3 × S 3 × S 1 with mixed R–R and NS–NS fields. We focus on strings ending along a straight line at the boundary of AdS3, which can be interpreted as line operators in a dual CFT2. We study both classical configurations and quadratic fluctuations around them. We find that strings sitting at a fixed point in S 3 × S 3 × S 1, i.e. satisfying Dirichlet boundary conditions, are 1/2 BPS. We also show that strings sitting at different points of certain submanifolds of S 3 × S 3 × S 1 can still share some fraction of the supersymmetry. This allows to define supersymmetric smeared configurations by the superposition of them, which range from 1/2 BPS to 1/8 BPS. In addition to the smeared configurations, there are as well 1/4 BPS and 1/8 BPS strings satisfying Neumann boundary conditions. All these supersymmetric strings are shown to be connected by a network of interpolating BPS boundary conditions. Our study reveals the existence of a rich moduli of supersymmetric open string configurations, for which the appearance of massless fermionic fields in the spectrum of quadratic fluctuations is crucial.

Scalar Casimir effect for a conducting cylinder in a Lorentz-violating background

Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green’s functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal–normal component of the stress–energy tensor) as a suitable second-order differential operator acting on the corresponding Green’s functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force.

Ensemble dependence of critical Casimir forces in films with Dirichlet boundary conditions.

In a recent study [Phys. Rev. E 94, 022103 (2016)2470-004510.1103/PhysRevE.94.022103] it has been shown that, for a fluid film subject to critical adsorption, the resulting critical Casimir force (CCF) may significantly depend on the thermodynamic ensemble. Here we extend that study by considering fluid films within the so-called ordinary surface universality class. We focus on mean-field theory, within which the order parameter (OP) profile satisfies Dirichlet boundary conditions and produces a nontrivial CCF in the presence of external bulk fields or, respectively, a nonzero total order parameter within the film. Additionally, we study the influence of fluctuations by means of Monte Carlo simulations of the three-dimensional Ising model. We show that, in the canonical ensemble, i.e., when fixing the so-called total mass within the film, the CCF is repulsive for large absolute values of the total OP, instead of attractive as in the grand canonical ensemble. Based on the Landau-Ginzburg free energy, we furthermore obtain analytic expressions for the order parameter profiles and analyze the relation between the total mass in the film and the external bulk field.

Spontaneous Radiation of a Two-Level System Confined in a Reflective Spherical Shell Quantum Dot

Using a first-order time-dependent perturbation theory, we calculate the spontaneous emission rate of a two-level system trapped between perfectly reflecting concentric spheres. The emitter is represented by a two-level monopole coupled to a Hermitian massless scalar field satisfying Dirichlet boundary conditions in such quantum-confined low-dimensional structure. We obtained the appropriate Green’s function evaluated in worldline of the atom which incorporates contributions from an infinite set of variable image charges. We provide an analytical expression for the decay rate to investigate the radiation process of the trapped atomic system. We perform a broad analysis of the dependence of the decay rate for different relations between the radii of spheres and the emitted radiation energy. We unveil regimes of strong suppression of the spontaneous emission rate as well as the development of irregular oscillations as a function of the quantum of emitted energy.

S-shaped bifurcation diagrams in exterior domains

We study a nonlinear eigenvalue problem on the exterior to a simply connected bounded domain in \(\mathbb {R}^N\) containing the origin. We consider positive weak solutions satisfying Dirichlet boundary conditions on the compact boundary and decaying to zero at infinity. We discuss multiplicity and uniqueness results of solutions with respect to a bifurcation parameter and conjecture an S-shaped bifurcation diagram for positive reaction terms which are singular at the origin and sublinear at infinity. As a by-product, on regions exterior to a ball with radially symmetric weight functions, we obtain radial symmetry of solutions when uniqueness holds.

The Helfrich boundary value problem

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower-semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Gro{\ss}e-Brauckmann.

Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

Abstract Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

The Casimir Effect from the Point of View of Algebraic Quantum Field Theory

We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ∗-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincare vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.

Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

A wide range of reaction–diffusion systems with constant diffusivities that are invariant under Q-conditional operators is found. Using the symmetries obtained, the reductions of the corresponding systems to the systems of ODEs are conducted in order to find exact solutions. In particular, the solutions of some reaction–diffusion systems of the Lotka–Volterra type in an explicit form and satisfying Dirichlet boundary conditions are obtained. An biological interpretation is presented in order to show that two different types of interaction between biological species can be described.

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

We study the existence of nontrivial, nonnegative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray–Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.

Spherical Casimir Effect for a Massive Scalar Field on the Three Dimensional Ball

The zeta function regularization technique is used to study the Casimir effect for a scalar field of mass $m$ satisfying Dirichlet boundary conditions on a spherical surface of radius $a$. In the case of large scalar mass, $ma\gg1$, simple analytic expressions are obtained for the zeta function and Casimir energy of the scalar field when it is confined inside the spherical surface, and when it is confined outside the spherical surface. In both cases the Casimir energy is exact up to order $a^{-2}m^{-1}$ and contains the expected divergencies, which can be eliminated using the well established renormalization procedure for the spherical Casimir effect. The case of a scalar field present in both the interior and exterior region is also examined and, for $ma\gg 1$, the zeta function, the Casimir energy, and the Casimir force are obtained. The obtained Casimir energy and force are exact up to order $a^{-2}m^{-1}$ and $a^{-3}m^{-1}$ respectively. In this scenario both energy and force are finite and do not need to be renormalized, and the force is found to produce an outward pressure on the spherical surface.

Non-equilibrium dynamics and AdS 4 Robinson-Trautman

The Robinson-Trautman space-times provide solutions of Einstein's equations with negative cosmological constant, which settle to $AdS_4$ Schwarzschild black hole at late times. Via gauge/gravity duality they should describe a system out of equilibrium that evolves towards thermalization. We show that the area of the past apparent horizon of these space-times satisfies a generalized Penrose inequality and we formulate as well as provide evidence for a suitable generalization of Thorne's hoop conjecture. We also compute the holographic energy-momentum tensor and deduce its late time behavior. It turns out that the complete non-equilibrium process on the boundary is governed by Calabi's flow on $S^2$. Upon linearization, only special modes that arise as supersymmetric zero energy states of an associated supersymmetric quantum mechanics problem contribute to the solution. We find that each pole of radiation has an effective viscosity given by the eigenvalues of the Laplace operator on $S^2$ and there is an apparent violation of the KSS bound on $\eta / s$ for the low lying harmonics of large $AdS_4$ black holes. These modes, however, do not satisfy Dirichlet boundary conditions, they are out-going and they do not appear to have a Kruskal extension across the future horizon ${\cal H}^+$.